Abstract

An analog computer makes use of continuously changeable quantities of a system, such as its electrical, mechanical, or hydraulic properties, to solve a given problem. While these devices are usually computationally more powerful than their digital counterparts, they suffer from analog noise which does not allow to error control. We will focus on analog computers based on active electrical networks comprised of resistors, capacitors, and operational amplifiers which are capable of simulating any linear ordinary differential equation. However, the class of nonlinear dynamics they can solve is limited. In this work, by adding memristors to the electrical network, we show that the analog computer can simulate a large variety of linear and nonlinear integro-differential equations by carefully choosing the conductance and the dynamics of the memristor state variable. We study the performance of these analog computers by simulating integro-differential models of fluid dynamics type, nonlinear Volterra equations for population growth, and quantum models describing non-Markovian memory effects, among others. Finally, we perform stability tests by considering imperfect analog components, obtaining robust solutions with up to 13% relative error for relevant timescales.

Introduction.— Analog computers employ continuously tunable quantities of a system, such as its electrical, mechanical, or hydraulic properties, to codify and solve a given problem. Usually, the dynamics of an analog computer perfectly matches the dynamics of the simulated system. The advantage of analog computers lies on the computational power provided by real-time operation and complete parallelism, so that they require less resources than a digital counterpart for the simulation. On the other hand, the accuracy of an analog computer is limited by its computing elements and by the quality of its analog components.
Despite the astonishing evolution of digital computers during the last decades, the interest in analog computers has re-emerged in recent years MacLennan (2014). For example, posible applications as math co-processors in VLSI architectures Cowan et al. (2005, 2006), and wave-based analog computation in metamaterials have recently been proposed Silva et al. (2014); AbdollahRamezani et al. (2015); Youssefi et al. (2016).

An electrical analog computer is an active network composed of electrical elements, namely, resistors, capacitors and operational amplifiers which, connected together, are capable of simulating any set of linear ordinary differential equations Kodali (1967). In this case, the solutions of these equations are encoded into the time evolution of the voltage waveform produced by the analog computer. Indeed, electrical analog devices can only work with a single independent variable when codified in time Pour-El (1974); Rubel (1988). Nonetheless, by making use of finite-difference methods, it is also possible to solve partial differential equations Kodali (1967); Ratier (2012).

Memristors, resistors whose resistance depends on the history of charges crossing through the device, are passive circuit elements theoretically postulated by Leon Chua Chua (1971); Chua and Kang (1976) and firstly constructed in HP Labs in a TiO2 thin film Strukov et al. (2008). Afterwards, various other physical implementations of memristors have been reported Jo et al. (2010); Chang et al. (2011); Chanthbouala et al. (2012); Kim et al. (2012). The memory and non-volatility properties of the memristor Chua (2011) have sparked great interest in its applications in the fabrication of digital memories Waser et al. (2009), and performance of computational tasks Borghetti et al. (2010); Pershin and Ventra (2012); Yang et al. (2012); Kulkarni and Teuscher (2012); Cassinerio et al. (2013). While most of the interest in memristors lies in applications in digital computation, there has also been proposals of utilization of memristors in programmable analog components of electrical networks Pershin and Ventra (2010) and in the design of highly efficient operational amplifiers Jahromi et al. (2017). Therefore, a natural question is to study whether the characteristic properties of memristors could extend the applications of eletrical analog computers. To the best of our knowledge, the role of memristors for efficient analog simulation of physical and mathematical models has not yet been explored.

In this work, by adding classical memristors to the network of an electrical analog computer, we show that it can simulate a large variety of linear and nonlinear integro-differential equations with interest in mathematics and physics. This is enabled by appropriately choosing the conductance of the memristors and the dynamics of its state variable. We study the performance of these analog computers by simulating integro-differential models of fluid dynamics, nonlinear Volterra equations for population growth, and quantum models describing non-Markovian memory effects, among others. Finally, by considering 10% error in the analog components, we perform stability tests of the dynamics, showing the robustness of the simulations with up to 13% error for relevant timescales. It is noteworthy to mention that the operation conditions of an off-the-shelf memristor fit those of usual circuit toolboxes

We will describe briefly the implementation of the aforementioned operations for the analog computer. First, let us consider the summation and sign inversion operations, which are enabled by an adder circuit whose transfer characteristics for n inputs are given by

Vout=−n∑iKiViin,

(1)

where Vout is the output voltage and Ki=Rf/Ri. The symbolic representation and equivalent circuit of the adder is shown in Fig. 1(a). Sign inversion can be implemented by an adder with a single input with K1=1.

Figure 1: Symbolic representation and corresponding circuit of the computing elements. (a) Adder for n inputs. (b) Integrator for n inputs. After the initial charge of the capacitor, that part of the circuit is switched off from the rest.

Next, integration, is meant to be the most important operation in an analog computer. It can be implemented by an integrator circuit, whose transfer characteristics for n inputs are given by

Vout(t)=−1C∫t0∑iViin(τ)Ridτ+IC,

(2)

where the constant IC stands for the initial condition. The symbolic representation and equivalent circuit of the integrator is shown in Fig. 1(b). For the adder and integrator circuits, we consider ideal operational amplifiers.

The last linear operation, multiplication by a constant, can be implemented with a potentiometer, which is described by

Vout=αVin,

(3)

where α<1 is a constant that characterizes the device.

In addition, analog computers can also simulate nonlinear dynamics to some extent, by considering signal generators and function multipliers, which enable the mathematical multiplication of two signals.

We will consider a simple example to illustrate the configuration of an analog computer for the simulation of a linear ordinary differential equation. Let us consider the following second-order linear differential equation

¨y(t)=−˙y(t)+y(t).

(4)

This equation is written in this form to remark that ¨y can be obtained as the sum of −˙y and y, which are obtained from integrating ¨y once and twice, respectively. The computer diagram that implements Eq. (4) is shown in Fig. 2. Therefore, we see that integrator circuits enable the solution of ordinary differential equations by making use of feedback connections, which make the input and output voltage in Eq. (2) to be equal. In addition, several integrators connected in series stand for subsequent integrations of the desired variable. In this way, the simulation of a linear differential equation of order n would require n integrators.

This method can be extended to a system of m linear ordinary differential equations of order n, which can be written as

m∑ℓ=1n∑k=0aℓ,kd(k)yℓ(t)dtk=0.

(5)

The simulation of such a system requires nm integrators, and since the summation can be absorbed into the integrators, it requires at most nm2 sign inverters.

It is important to note that in order to solve a differential equation with an analog computer it is not necessary to know the voltage waveform at any point of the electrical network, provided that the circuit was properly constructed, the desired solution is obtained by measuring the voltage signal at the output of the circuit.

Memristive analog computation.— A voltage-controlled memristor can be described by the following current-voltage relation Chua (1971)

i

=

g(ω)v,

(6)

˙ω

=

f(v),

(7)

where i(t) and v(t) are the current and voltage across the device, respectively. The quantity ω is an internal state variable specified by the physical implementation of the memristor. For ideal voltage-controlled memristors the state variable corresponds to the magnetic flux ϕ, which means f(v)=v. The quantity g(ω) is the so-called memductance which has units of conductance and is a function of the state variable. Since the memristor is a passive circuit element, the memductance g(ω) must always be positive.

The notion of memristive behavior can be extended to a broader class of system whose characteristics resemble those of the memristor, and are called memristive systems Chua and Kang (1976). These are defined by

i

=

g(ω,v,t)v,

(8)

˙ω

=

f(ω,v,t).

(9)

Where in general, ω represents a set of n state variables, and f and g are continuous functions.

Now, to study the inclusion of memristive devices into an analog computer we will consider the case of an integrator circuit with a single input, where we substitute the resistor for a memristor, as is shown in Fig. 3. We will consider that the memristor has a single state variable ω. In the figure, the current across the memristor, iM, is given by Eq. (8). Then, by Kirchhoff’s law at node A, we have

which in general will be a nonlinear integro-differential equation that will be specified by g and f, which describe the memductance and the memristor state variable dynamics, respectively. It must be noticed that these functions are specified by memristor engineering.

Figure 3: Integrator circuit where we have replaced the resistor for a memristor. After the initial charge of the capacitor, the part denoted by IC is switched off from the rest of the circuit.

We now show the type of equations that can be solved using this circuit. Let us consider the equation that Volterra introduced for single populations in the study of growth of biological populations Volterra (2005),

˙N(t)=N(t)[a−bN(t)−∫t0k(t−s)N(s)ds],t∈R+,

(13)

in which N(t) is the population size at time t, a and b are positive rate constants, and k(t) is the “hereditary” influence.

Now, in order to simulate this equation, we consider a memristive system as described by Eq. (8) and Eq. (9). By assuming g(ω,v,t)=−a+bv+k1(t)ω and ˙ω=k2(t)v, then, substituting into Eq. (12) we have

˙v(t)=(a−bv(t)−∫t0k(t,τ)v(τ)dτ)v(t),

(14)

where we have chosen C=1 and k(t,τ)=k1(t)k2(τ). This is restricted, however, to kernels that can be separated in this way, otherwise it is only possible to simulate single-variable kernels. This equation can be implemented with a single integrator and memristor as shown in Fig. 3.

By connecting integrator circuits in series, we can generate nonlinear integro-differential equations of higher-order. For example, if we consider n integrators in series, where only the first integrator circuit contains a memristor, then we can generate the nth-order integro-differential equation

d(n)vdt(n)=−1Cg(∫t0f(ω,v,τ)dτ,v(t),t)v(t).

(15)

On the other hand, if each of the n integrators contains a memristor, then the nth-order integro-differential equation will involve the composition of n memductance functions g1,…,gn corresponding to each memristor. The resulting equation can be written as follows

Missing or unrecognized delimiter for \bigg

(16)

This allows for the simulation of a wide class of nonlinear integro-differential equation.

It is also possible to solve first-order linear integro-differential equations with a suitable change of variables. If we consider Eq. (12) and define the memristor by g(ω,v,t)=−ω and ˙ω=k(t)ln(v), we obtain

˙v(t)=1C(∫t0k(s)ln(v(s))ds)v(t).

(17)

Then, with u=ln(v), we have

˙u(t)=1C(∫t0k(s)u(s)ds).

(18)

Notice that we can avoid the sign inversion of the integrand by normalizing the voltage such that 0≤v≤1. In this way, it is possible to simulate several linear integro-differential equations coming from quantum models. An example of this is a Volterra equation describing non-Markovian quantum memory effects Alvarez-Rodriguez et al. (2017), given by

∂tρ=∫t0K(t,s)Lρ.

(19)

Since L is a linear superoperator, the integral kernel will involve linear terms in the elements of ρ. This leads to a system of integro-differential equations for the elements of ρ which can be implemented by appropriately choosing the dynamics of the internal state variable of the memristor.

Figure 4: Stability test of the solution of (a) Eq. (22) and (b) Eq. (23). (Left) Shows the exact solution (black solid line) and the solution considering imperfect analog components (red dashed line) with up to 10% error in their corresponding coefficients. (Right) Shows the average relative error over 100 iterations of the simulation with imperfect analog components.

Aditionally, linear integro-differential equations that appear in the model of turbulent diffusion Velikson (1975), have the form

˙u(t)+p(t)u(t)+∫t0K(t,s)u(t−s)u(s)ds=0.

(20)

Equations of this type can be implemented with a memristor defined by g(ω,v,t)=α(t)k1(t)ω and ˙ω=k2(t)α(t)2ln(v)2, such that K(t,s)=k1(t)k2(s), and where α(t)=exp(∫t0p(s)ds). By using z(t)=α(t)u(t), we can configure Eq. (20) into the analog computer as the following equation,

˙z(t)=−(∫t0k′(s)z(s)2ds),

(21)

with k′(s)=k(s)α(s)2. Next, connecting the output z(t) to a signal multiplier with the signal 1α(t), we recover the solution u(t) of Eq. (20).

In order to provide an example of the previous procedures and to test the stability of the solutions, let us consider the following integro-differential equation

˙N(t)=2N(t)−0.001N(t)2−N(t)∫t0s1+se−(t−s)N(s)ds.

(22)

This equation can be implemented by taking C=1 and chosing g(ω,v,t)=−2+0.001v+e−tω and ˙ω=ess1+sv in Eq. (12). In Fig. 4(a), we show the exact solution of this equation compared against the case with up to 10% error in the coefficients of Eq. (22), which stands for imperfections in the analog components of the computer. We can see that the average relative error stabilizes at ∼10% indicating that the solution is robust against imperfections in the analog components.

As a second example we consider the following integro-differential equation

˙u(t)=−(18e−2tu(t)+∫t012e−(t+s)u(s)2ds).

(23)

This equation can be taken into the form of Eq. (21) by considering α(t)=exp(−1/16(e−2t−1)), k1(t)=1/2e−t, k2(s)=e−s and ˙ω=k2(s)ln(v)2.
In Fig. 4(b), we show the solution of this equation compared against the case with up to 10% error in the coefficients of Eq. (23), as well. We observe that, in this case, the error increases continuously, as it is usually the case in analog computers, reaching up to 13% within the timescale considered.

Conclusions.—
We have studied the inclusion of memristors into the network of electric analog computers. We have found that this addition enables analog computers to simulate linear and nonlinear integro-differential equations by appropriately choosing the memductance and the dynamics of the memristor state variable. This broadens the applicability of analog computers which, otherwise, could only solve systems of ordinary differential equations. We study the performance of these analog computers by simulating integro-differential models of fluid dynamics, nonlinear Volterra equations for population growth, and quantum models describing non-Markovian memory effects. Additionally, we test the stability of the solutions against imperfections of analog components by introducing up to 10% error in their corresponding coefficients, finding that the relative error reaches up to 13% for relevant timescales. In this manner, we show the robustness of the introduced methods. Moreover, it is noteworthy to mention that our results have been obtained with minimal architecture variations, leaving the possibility of complex arrangements open for future studies. As a further scope, it would be interesting to consider the proposed scenarios in the context of quantum memristors Pfeiffer et al. (2016); Shevchenko et al. (2016); Salmilehto et al. (2017); Sanz et al. (2017) in the search of quantum improvements for analog computers.

The authors would like to thank Unai Alvarez-Rodriguez and Massimiliano Di Ventra for useful discussions.
G.A.B. acknowledges support from CONICYT Doctorado Nacional 21140587 and Dirección de Postgrado USACH. J.C.R. thanks FONDECYT for support under grant No. 1140194. While M.S. and E.S. are grateful for the funding of Spanish MINECO/FEDER FIS2015-69983-P and Basque Government IT986-16. This material is also based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advance Scientific Computing Research (ASCR), under field work proposal number ERKJ335.